Pseudo-Anosov maps and pairs of filling simple closed geodesics on Riemann surfaces

TL;DR

This paper investigates the filling properties of pairs of simple closed geodesics on punctured Riemann surfaces under pseudo-Anosov maps, revealing conditions under which these pairs fill the surface and characterizing special cases.

Contribution

It establishes new filling criteria for pairs of geodesics under pseudo-Anosov maps and characterizes the unique geodesic disjoint from both in specific cases.

Findings

For any pseudo-Anosov map isotopic to the identity on the surface with puncture, (a, f^m(a)) fills the surface for m ≥ 3.

If (a, f^2(a)) does not fill, there is a unique geodesic b disjoint from both, which is f(a).

When a and f(a) are not disjoint, then (a, f^m(a)) fills the surface for all m ≥ 2.

Abstract

Let $S$ be a Riemann surface with a puncture $x$. Let $a\subset S$ be a simple closed geodesic. In this paper, we show that for any pseudo-Anosov map $f$ of $S$ that is isotopic to the identity on $S\cup \{x\}$, $(a, f^m(a))$ fills $S$ for $m\geq 3$. We also study the cases of $0<m\leq 2$ and show that if $(a,f^2(a))$ does not fill $S$, then there is only one geodesic $b$ such that $b$ is disjoint from both $a$ and $f^2(a)$. In fact, $b=f(a)$ and $\{a,f(a)\}$ forms the boundary of an $x$-punctured cylinder on $S$. As a consequence, we show that if $a$ and $f(a)$ are not disjoint. Then $(a,f^m(a))$ for any $m\geq 2$ fills $S$.